The quantity of rotational motion in a body is called its angular momentum. Thus angular momentum plays the same role in rotational motion as played by linear momentum in translational motion.

Mathematically, angular momentum is the cross-product of the position vector and the linear momentum, both measured in an inertial frame of reference.

ρ = r × P

The magnitude of the angular momentum vector is

ρ = r P sinθ (magnitude)

ρ = r m V sinθ (since P = m V)

where, V is linear speed, θ is the angle between r and P

θ= 90^{°} in circular motion (special case)

The direction of the angular momentum can be determined by the Right-Hand Rule.

Also ρ = r m (r ω) sinθ

ρ = m (r^{2} ω) sinθ

**Units of Angular Momentum**

The **units of angular momentum** in S.I system are kgm^{2}/s or Js.

- ρ = r m V sinθ

= m × kg × m/s

= kg.m^{2}/s

- ρ = r P sinθ

= m × Ns

= (Nm) × s

= J.s

Dimensions of Angular Momentum

[ρ] = [r][P]

= [r] [m] [V]

= L.M.L/T

= L^{2} M T^{-1}

**Relation between Torque and Angular Momentum**

**Or **

**Prove that the rate of change of angular momentum is equal to the external torque acting on the body.**

**Proof:- **We know that rate of change of linear momentum is equal to the applied force.

F = ma

Where P = mv is a linear momentum of the particle.

Taking the vector product of r with both sides, we get

But r × F = , Therefore, we have

= r × (since τ = r × P)

Now, according to the definition of angular momentum

ρ = r × P

Taking derivative w.r.t time, we get

Or, Rate of change of Angular Momentum = External Torque **(Proved)**.

## Conservation of Angular Momentum

According to the law of conservation of angular momentum, if the external torque (couple) acting on a system is zero, the total angular momentum of a rigid body or a system of particles is conserved.

If the moment of inertia of the body changes from I_{1} to I_{2} due to the change of the distribution of mass of the body, then angular velocity of the body changes from ω1 to ω2, such that

If τ_{ext }0, then L = I(ω) = constant ⇒ I_{1}ω_{1} = I_{2}ω_{2}

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